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Self-similarity relations for cooling superfluid neutron stars

We consider models of cooling neutron stars with nucleon cores which possess moderately strong triplet-state superfluidity of neutrons. When the internal temperature drops below the maximum of the critical temperature over the core, $T_{\rm C}$, this superfluidity sets in. It produces a neutrino outburst due to Cooper pairing of neutrons which greatly accelerates the cooling. We show that the cooling of the star with internal temperature $T$ within $0.6\,T_{\rm C} \lesssim T \leq T_{\rm C}$ is described by analytic self-similar relations. A measurement of the effective surface temperature of the star and its decline, supplemented by assumptions on star's mass, radius and composition of heat-blanketing envelope, allows one to construct a family of cooling models parametrized by the value of $T_{\rm C}$. Each model reconstructs cooling history of the star including its neutrino emission level before neutron superfluidity onset and the intensity of Cooper pairing neutrinos. The results are applied to interpret the observations of the neutron star in the Cassiopeia A supernova remnant.